Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated.     

Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces

This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.     

Well-posedness and regularity of backward stochastic Volterra integral equations

Backward stochastic Volterra integral equations (BSVIEs, for short) are studied. Notion of adapted M-solution is introduced. Well-posedness of BSVIEs is established and some regularity results are proved for the adapted M-solutions via Malliavin calculus. A Pontryagin type maximum principle is presented for optimal controls of stochastic Volterra integral equations.     

Strong Convergence of the Euler-Maruyama Method for Nonlinear Stochastic Volterra Integral Equations with Time-Dependent Delay

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper. Strong convergence rate (at fixed point) of the corresponding Euler-Maruyama method is obtained when coefficients $f$ and $g$ both satisfy local Lipschitz and linear growth conditions. An example is provided to interpret our conclusions. Our result generalizes and improves the conclusion in [J. Gao, H. Liang, S. Ma, Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay, Appl. Math. Comput., 348 (2019) 385-398.]     

Regularity of solutions to stochastic Volterra equations with infinite delay

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a -dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.

     

Backward stochastic Volterra integral equations and some related problems

Backward stochastic Volterra integral equations (BSVIEs, for short) are introduced. The existence and uniqueness of adapted solutions are established. A duality principle between linear BSVIEs and (forward) stochastic Volterra integral equations is obtained. As applications of the duality principle, a comparison theorem is proved for the adapted solutions of BSVIEs, and a Pontryagin type maximum principle is established for an optimal control of stochastic integral equations.     

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

In this paper, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation. Then, the large deviation estimates of small perturbation to equations of this type are obtained. We finally apply them to SDEs with the kernel of fractional Brownian motion with Hurst parameter H∈(0,1).     

Time regularity for stochastic Volterra equations by the dilation theorem

The dilation theorem of Nagy is applied to establish time regularity of the solutions to a class of stochastic evolutionary Volterra equations.     

Linear quadratic stochastic integral games and related topics

This paper studies linear quadratic games problem for stochastic Volterra integral equations(SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways.As a consequence,the open problems raised by Chen and Yong(2007) are solved.To characterize the saddle points more clearly,coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced.Compared with deterministic game problems,some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems.Moreover,our representations and arguments are even new in the classical SDEs case.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Remarks on stochastic permanence of population models

This paper examines the asymptotic behavior of the stochastic Lotka–Volterra model under Markovian switching. We show that the stochastic Lotka–Volterra model is stochastically permanent. Moreover, we give another type of stochastic permanence, and consider the relationship of two types of stochastic permanence under white noise perturbation.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Strong Convergence of the Euler-Maruyama Method for a Class of Stochastic Volterra Integral Equations

In this paper, we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations (SVIEs). It is known that the strong convergence order of the Euler-Maruyama method is $\frac12$. However, the strong superconvergence order $1$ can be obtained for a class of SVIEs if the kernels $\sigma_{i}(t, t) = 0$ for $i=1$ and $2$; otherwise, the strong convergence order is $\frac12$. Moreover, the theoretical results are illustrated by some numerical examples.     

A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.     

Continuous-time dynamic risk measures by backward stochastic Volterra integral equations

Continuous-time dynamic convex and coherent risk measures are introduced. To obtain existence of such risk measures, backward stochastic Volterra integral equations (BSVIEs, for short) are studied. For such equations, notion of adapted M-solution is introduced, well-posedness is established, duality principles and comparison theorems are presented. Then a class of dynamic convex and coherent risk measures are identified as a component of the adapted M-solutions to certain BSVIEs.     

Economical Runge-Kutta methods with strong global order one for stochastic differential equations

Economical Runge-Kutta schemes for the numerical solution of Stratonovich stochastic differential equations are proposed. The methods have strong global order 1. Numerical stability is studied and some examples are presented to support the theoretical results.     

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??In this paper, we study a class of stochastic Volterra equations, which include the stochastic differential equation driven by fractional Brownian motion. By using a maximal inequality due to It\^o (1979), we establish the central limit theorem for stochastic Volterra equation on the continuous path space, with respect to the uniform norm.     

Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane

In this paper we study stochastic Volterra equations in a plane. These equations contain integrals with respect to fields of locally bounded variation and square-integrable strong martingales. We prove the existence and the uniqueness of solutions of such equations with locally integrable (in some measure) trajectories, assuming that the coefficients of equations possess the Lipschitz property with respect to the functional argument. We prove that a solution of a stochastic Volterra integral equation in a plane is continuous with respect to parameter.     

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.     

Backward stochastic Volterra integral equations—Representation of adapted solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward–backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.